Main variables:

When we write \(a\) instead of \(\vec{a}\), we refer to the magnitude of that vector. For instance, \(\vec{a}\) is the acceleration vector, and \(a\) is the magnitude of the acceleration.

Main Equations:

The dot product between two vectors:

\begin{equation*}
\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y,
\end{equation*}

or

\begin{equation*}
\vec{A} \cdot \vec{B} = A B \cos \theta,
\end{equation*}

where \( \theta \) is the angle between the two vectors.

The magnetic force \(\vec{F}\) (Lorentz Law):

\begin{equation*}
\vec{F} = q \vec{v} \times \vec{B}.
\end{equation*}

The radius of the circular trajectory for a charged particle in a magnetic field is:

\begin{equation*}
R = \frac{m v}{|q|\vec{B}}.
\end{equation*}

The magnetic force for a wire carrying current \(I\) is:

\begin{equation*}
\vec{F} = I \vec{\ell} \times \vec{B}.
\end{equation*}

Differential form of Biot and Savart law:

\begin{equation*}
d \vec{B} = \frac{\mu_0}{4 \pi} \frac{I d\vec{\ell} \times \hat{\textbf{r}} }{r^2}.
\end{equation*}

Magnetic field of a circular loop:

\begin{equation*}
B = \frac{\mu_0 I a^2}{2 (x^2 + a^2)^{3/2} }.
\end{equation*}

Magnetic torque:

\begin{equation*}
\vec{\tau} = \vec{\mu} \times \vec{B}.
\end{equation*}

Potential energy for a magnetic dipole:

\begin{equation*}
U = -\vec{\mu} \cdot \vec{B}.
\end{equation*}

Ampere’s law:

\begin{equation*}
\oint \vec{B} \cdot d \vec{\ell} = \mu_0 I_{\text{enc}}.
\end{equation*}

Magnetic field near a long straight current-carrying conductor:

\begin{equation*}
B = \frac{\mu_0 I}{2 \pi R}.
\end{equation*}

Force between parallel conductors (attractive force):

\begin{equation*}
F = \frac{\mu_0 I I’}{2 \pi R}.
\end{equation*}

Force between antiparallel conductors (repulsive force):

\begin{equation*}
F = -\frac{\mu_0 I I’}{2 \pi R}.
\end{equation*}

Magnetic flux:

\begin{equation*}
\Phi_B = \vec{B} \cdot \vec{A}.
\end{equation*}

Magnetic flux in integral form:

\begin{equation*}
\Phi_B = \int \vec{B} \cdot d \vec{A}.
\end{equation*}

Faraday’s law:

\begin{equation*}
\varepsilon = – \frac{\Delta \Phi_B}{ \Delta t}.
\end{equation*}

Faraday’s law in differential form:

\begin{equation*}
\varepsilon = – \frac{d \Phi_B}{dt}.
\end{equation*}

Lenz’s law:

• If the magnetic flux increases, the induced magnetic field opposes the original one.
• If the magnetic flux decreases, the induced magnetic field points in the same direction as the original one.